0,2

Let u(k), v(k), w(k) be the 3 sequences defined by u(1)=1, v(1)=0, w(1)=0 and u(k+1)=u(k)+v(k)-w(k), v(k+1)=u(k)-v(k)+w(k), w(k+1)=-u(k)+v(k)+w(k); let M(k)=Max(u(k),v(k),w(k)); then a(n)=M(2n)=M(2n-1) - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 25 2002

Also the number of words of length 2n generated by the two letters s and t that reduce to the identity 1 by using the relations ssssss=1, tt=1 and stst=1. The generators s and t along with the three relations generate the dihedral group D6=C2xD3. - Jamaine Paddyfoot and John Layman (jay_paddyfoot(AT)hotmail.com/layman(AT)math.vt.edu), Jul 08 2002

Binomial transform of A025192. - Paul Barry (pbarry(AT)wit.ie), Apr 11 2003

a(n) = A020988(n-1)+1 = A039301(n+1)-1 = A083584(n-1)+2. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jun 14 2003

Number of walks of length 2n+1 between two adjacent vertices in the cycle graph C_6. Example: a(1)=3 because in the cycle ABCDEF we have three walks of length 3 between A and B: ABAB, ABCB, and AFAB. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 01 2004

a(n) = A072197(n) - A020988(n). - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Dec 31 2004

Numbers of the form 1+Sum_{i=1..m} [2^(2i-1)]. - Artur Jasinski (grafix(AT)csl.pl), Feb 09 2007

Prime numbers of the form 1+Sum[2^(2n-1)] are in A000979. Numbers x such 1+Sum[2^(2n-1)] is prime for n=1,2,...,x is A127936. - Artur Jasinski (grafix(AT)csl.pl), Feb 09 2007

Related to A024493(6n+1), A131708(6n+3), A024495(6n+5). - Paul Curtz (bpcrtz(AT)free.fr), Mar 27 2008

H. W. Gould, Combinatorial Identities, Morgantown, 1972, (1.77), page 10.

S. Hong and J. H. Kwak, Regular fourfold covering with respect to the identity automorphism, J. Graph Theory, 17 (1993), 621-627.

C. Bebeacua, T. Mansour, A. Postnikov and S. Severini, On the x-rays of permutations

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 893

Eric Weisstein's World of Mathematics, Repunit

a(n) = sum(A060920(n, m), m = 0..n) = A002450(n+1)-2*A002450(n). G.f.: (1-2*x)/(1-5*x+4*x^2). - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Apr 24 2001

a(n)=sum(binomial(n+k, 2*k)/2^(k-n), k=0..n). a(n)=4a(n-1)-1, n>0.

a(n)=1 + 2*sum{k=0..n-1, 4^k} a(n)=A001045(2n+1). - Paul Barry (pbarry(AT)wit.ie), Mar 17 2003

u(0) = 0; u(n+1) = 4*u(n) - 1 - Regis Decamps (decamps(AT)users.sf.net), Feb 04 2004

a(n)=sum(i+j+k=n, (n+k)!/i!/j!/(2*k)!) 0<=i, j, k<=n - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 25 2004

a(n)=5a(n-1)-4a(n-2). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 01 2004

a(n)=4^n-A001045(2n) - Paul Barry (pbarry(AT)wit.ie), Apr 17 2004

a(n)=2*(A001045(n))^2+(A001045(n+1))^2. - Paul Barry (pbarry(AT)wit.ie), Jul 15 2004

a(n) = left and right terms in M^n * [1 1 1] where M = the 3X3 matrix [1 1 1 / 1 3 1 / 1 1 1]. M^n * [1 1 1] = [a(n) A002450(n+1) a(n)] E.g. a(3) = 43 since M^n * [1 1 1] = [43 85 43] = [a(3) A002450(4) a(3)]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 18 2004

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=4*a[n-1]-1 od: seq(a[n], n=1..24); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 22 2008

a = {}; Do[c = 1 + Sum[2^(2n - 1), {n, 1, x}]; AppendTo[a, c], {x, 0, 30}]; a - Artur Jasinski (grafix(AT)csl.pl), Feb 09 2007

a = {}; ZZ = 1; Do[ZZ = ZZ + 4^(x); AppendTo[a, ZZ], {x, 0, 24}]; a/2 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 03 2007

(PARI) a(n)=sum(k=-n\3, n\3, binomial(2*n+1, n+1+3*k))

a(n) = (2*A002450(n))+1. Cf. also A006054, A006356, A005578.

Partial sums of A081294.

Cf. A002450.

Cf. A000979, A000978, A124400, A124401, A127955, A127956, A127957, A127958, A127936.

Adjacent sequences: A007580 A007581 A007582 this_sequence A007584 A007585 A007586

Sequence in context: A106876 A034477 A084643 this_sequence A026671 A026876 A059278

nonn,easy

Simon Plouffe (plouffe(AT)math.uqam.ca)